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Further results on iterative methods for computing generalized inverses. (English) Zbl 1190.65088

Let A be a complex Banach algebra with unit 1; 𝒳,𝒴 two complex Banach spaces and (𝒳,𝒴) the set of all bounded linear operators from 𝒳 to 𝒴. The main theorem is stated as follows: Define the sequence

X k =X k-1 +βY(I 𝒴 -AX k-1 ),k=1,2,,

where β{0} and X 0 (𝒴,𝒳) with YYAX 0 · Then the above iteration converges if and only if ρ(I 𝒳 -βYA)<1, equivalently, ρ(I 𝒴 -βAY)<1·

In this case suppose now (ρ(I 𝒳 -βYA)<1 and that T,S are closed subspaces of 𝒳,𝒴· If moreover (Y)=T,   𝒩(Y)=Sand (X 0 )T, then A T,S (2) exists and {X k } converges to A T,S (2) and if q=min(I 𝒳 -βYA,I 𝒴 -βAY)<1:

A T,S (2) -X k |β|q k 1-qYI 𝒴 -AX 0 ·

Here A T,S (2) denotes the generalized inverse. Several following theorems yield variants of the above theorem which state conditions under which the iterative procedures approximate the generalized inverse. The next section entitled “The generalized Drazin inverse of Banach algebra elements” defines an iteration approximating this inverse and different conditions under which this sequence converges to the Drazin inverse. Finally, the last section is devoted to a numerical example in which A 5×4 ·

65J10Equations with linear operators (numerical methods)
47A05General theory of linear operators
15A09Matrix inversion, generalized inverses
65F20Overdetermined systems, pseudoinverses (numerical linear algebra)